Automation Systems Lecture 8 - The quality of the control system. Jakub Mozaryn

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Automation Systems

Lecture 8 - The quality of the control system.

Jakub Mozaryn

Institute of Automatic Control and Robotics, Department of Mechatronics, WUT

Warszawa, 2016

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Quality of the control system

Appart from the most important requirement of asymptotic stability, there are imposed additional requirements on control systems, concer- ning the transient (dynamic) response and steady states. They are generally referred to as quality requirements of the control system.

The requirements related to the steady state are formulated by determining the so-called static accuracy of the control system - perimissible values of deviations of the system output from the set point in steady states (steady state errors) in the case of disturbances or set point changes.

Requirements related to the transient response in the control systems are determined by a number of indices, generally called dynamic quality in- dices of the control system.

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Quality of the control system

The task of the control system is to minimize the deviation from the set point, when time approaches infinity, described as an error in steady state:

e(t) = ez(t) + ew(t), (1)

where

ez(t) - error caused by disturbance,

ew(t) - error caused by change of the set point.

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Quality of the control system

When rating the quality of the control LTI system, because of the superposition property, both components of the steady state error e(t) = ez(t) + ew(t), can be analyzed separately.

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Steady state error caused by disturbance

Transfer function

Gz(s) = ∆ym(s)

z(s) =ez(s)

z(s) = ±Gz(s)Gob(s)

1 + G_ob(s)G_r(s) (2) e_z(s) = ∆y_m(s) = ±G_z(s)G_ob(s)

1 + Gob(s)Gr(s)· z(s) (3) Steady state error caused by disturbance

ezst.= lim

t→∞ez(t) = lim

s→0s · ez(s) (4)

ezst. = lim

s→0s · ±Gz(s)Gob(s)

1 + Gob(s)Gr(s)· z(s) (5)

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Steady state error caused by change of the set point

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Steady state error caused by change of the set point

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Steady state error caused by change of the set point

Transfer function

G_ew(s) = ew(s)

∆w (s) = −1

1 + G_ob(s)G_r(s) (6) ew(s) = −1

1 + Gob(s)Gr(s)∆w (s) (7) Steady state error caused by change of the set point

e_wst.= lim

t→∞e_w(t) = lim

s→0s · e_w(s) (8)

e_wst.= lim

s→0s · −1

1 + Gob(s)Gr(s)∆w (s) (9)

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Steady state error - example

Determine the steady state error of the control system shown in the figure, caused by step change of disturbances z(t) = 2 and step change of setpoint

∆w (t) = 5. Assume, that in the control system there is used:

P controller PD controller PI controller

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Steady state error - example

Transfer function

Gob(s) = kob

(Ts + 1)⁴ (10)

P controller

Gr(s) = kp (11)

PD controller

Gr(s) = kp(1 + Tds) (12) PI controller

Gr(s) = kp

1 + 1

Tis

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z(t) = 2 → z(s) = 2

s (14)

Change of the set point

∆w (t) = 5 → ∆w (s) =5

s (15)

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Steady state error caused by the disturbance - example, P controller

ezst.= lim

s→0s Gob(s)

1 + G_ob(s)G_r(s)z(s) (16) P controller

ezst.P = lims→0s Gob(s) 1 + Gob(s)Gr(s)

2 s =

lims→0

kob

(Ts + 1)⁴· 2 1 + kob

(Ts + 1)⁴kp

= lims→0

kob· 2 (Ts + 1)⁴+ k_ob· k_p

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Steady state error caused by the disturbance - P controller ezst.P = kob· 2

1 + kobkp

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Steady state error caused by the disturbance - example, PD controller

e_zst.= lim

t→∞e_z(t) = lim

s→0s Gob(s)

1 + G_ob(s)G_r(s)z(s) (19) PD controller

ezst.PD = lims→0

kob

(Ts + 1)⁴· 2 1 + kob

(Ts + 1)⁴kp(1 + Tds)

=

= lims→0

kob· 2

(Ts + 1)⁴+ k_ob· kp(1 + T_ds)

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Steady state error caused by the disturbance - PD controller ezst.PD= k_ob· 2

1 + kobkp

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Steady state error caused by the disturbance - example, PI controller

ezst.= lim

s→0s Gob(s)

1 + Gob(s)Gr(s)z(s) (22) PI controller

e_zst.PI = lim_s→0

k_ob (Ts + 1)⁴· 2 1 + kob

(Ts + 1)⁴kp(1 + 1 T_is)

=

= lims→0

k_ob· 2

(Ts + 1)⁴+ kob· kp(1 + 1 Tis)

= 0

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Steady state error caused by the disturbance - PI controller

ezst.PI = 0 (24)

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Steady state error caused by the disturbance - summary

P controller

e_zst.P = k_ob· 2 1 + kobkp

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PD controller

ezst.PD= kob· 2 1 + kobkp

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PI controller

e_zst.PI = 0 (27)

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Steady state error caused by change of the set point - example, P controller

ewst.= lim

s→0s −1

1 + G_ob(s)G_r(s)∆w (s) (28) P controller

e_wst.P = lim

s→0s −1

1 + Gob(s)kp

5 s = lim

s→0

−5 1 + kob

(Ts + 1)⁴kp

= −5

1 + kobkp

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Steady state error caused by change of the set point - P controller ewst.P = −5

1 + kobkp

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Steady state error caused by change of the set point - example, PD controller

ewst.= lim

s→0s −1

1 + G_ob(s)G_r(s)∆w (s) (31) PD controller

e_wst.PD = lim_s→0s −1

1 + G_ob(s)k_p(1 + T_ds) 5 s

= lims→0

−5 1 + k_ob

(Ts + 1)⁴k_p(1 + T_ds)

= −5

1 + kobkp

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Steady state error caused by change of the set point - PD controller ewst.PD = −5

1 + kobkp

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Steady state error caused by change of the set point - example, PI controller

e_wst.= lim

t→∞e_z(t) = lim

s→0s −1

1 + Gob(s)Gr(s)∆w (s) (34) PI controller

e_wst.PI = lim_s→0s −1 1 + G_ob(s)k_p

1 + 1

Tis

5 s

= lims→0

−5 1 + k_ob

(Ts + 1)⁴kp

1 + 1

Tis

= 0

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Steady state error caused by change of the set point - PI controller

ewst.PI = 0 (36)

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Steady state error caused by change of the set point - summary

P controller

e_wst.P = −5 1 + kobkp

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PD controller

e_wst.PD = −5 1 + kobkp

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PI controller

ewst.PI = 0 (39)

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Conclusions about steady state errors

In a control system with a static object and P or PD control algorithm there are non-zero steady state errors in relation to the disturbances or setpoint changes respectively.

Increasing the proportional gain of P or PD controller reduces the value of static deviations. Reducing the static deviation by increasing the gain k_p is usually limited due to the stability of the system. (The system with PD controller reaches the border of stability at higher gain than in the case of the regulator P.).

Integral action in the controller (PI, PID) provides zero steady state errors in relation to the disturbances or setpoint changes respectively.

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Dynamical quality of control system

Requirements related to the transient response in the control systems are determined by a number of indices, generally called dynamic

performance quality indicies of the control system. Groups of such indices are:

transient response indices,

indices descibing the frequency plots of the control system - magnitude and phase margins,

integral indices.

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Transient response indices

To evaluate the transient response following indices are used:

Maximum error (dynamical): e_m - the maximum value of error after the step change of disturbance or setpoint.

Settling time: t_r - it is the time between the moment of change of the set point w (t), or introduction of disturbances z(t) , and the moment when the error e(t) reaches a fixed value inside a boundary

∆e(t) (eg.∆e(t) = |0.05emax|).

Overshoot:

κ =

e2

e₁

· 100% (40)

where e1 and e2are the first 2 consecutive biggest errors with opposite signs, assuming steady state value of output y (t) after transient response as the zero level (baseline).

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Oscillatory transient response - disturbances

Rysunek :Oscillatory transient response of the control system to disturbances:

a) with non-zero static deviation, b) with zero static deviation

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Aperiodic transient response - disturbances

Rysunek :Aperiodic transient response of the control system to disturbances:

a) with non-zero steady state error, b) with zero steady state error

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Oscillatory transient response - setpoint

Rysunek :Oscillatory transient response of the control system to setpoint change: a) with non-zero steady state error, b) with zero steady state error

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Aperiodic transient response - setpoint

Rysunek :Aperiodic transient response of the control system to setpoint change: a) with non-zero steady state error, b) with zero steady state error

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